> print("Descriptive Statistics for RT across all Trials")
[1] "Descriptive Statistics for RT across all Trials"
> univSummary(data = Example6b, vars = "RT")

 Variable    N    Mean      SD  Variance     Min      Max  Median
 RT       1392 632.383 146.083 21340.288 352.000 1806.000 604.000

> print("Number of Items per Item Predictor Level")
[1] "Number of Items per Item Predictor Level"
> freqSummary(data = Example6b[!duplicated(Example6b$ItemID), ], 
+ vars = c("freq01", "size01"))

 Variable Level  N Proportion
 freq01   0     20      0.513
 freq01   1     19      0.487
 size01   0     20      0.513
 size01   1     19      0.487

> table(Example6b$freq01[!duplicated(Example6b$ItemID)], Example6b$size01[!duplicated(Example6b$ItemID)])
   
     0  1
  0 10 10
  1 10  9
> print("Model 1: Empty Means, Random Intercept for Subjects Only")
[1] "Model 1: Empty Means, Random Intercept for Subjects Only"
> Model1 = lmer(data = Example6b, REML = TRUE, formula = RT ~ 1 + 
+ (1 | SubjectID))
> lmerSummary(Model1, ICC1 = TRUE, LRT = TRUE, CI = TRUE)

Model Fit for Predicting RT:
      AIC       BIC    logLik -2*logLik 
17546.265 17561.980 -8770.132 17540.265 

Model for the Variance:
 Groups     Name       Variance Std.Dev
 SubjectID  Intercept  5167.096  71.883
 Residual             16307.206 127.700
Number of obs: 1392 , groups: SubjectID, 38 

Fixed Effects (with 95% CI):
 Term           Est     SE     DF t.value p.value   Lower   Upper
 Intercept  631.417 12.154 37.004  51.951   <.001 606.790 656.043

ICC1 intraclass correlation:
    Adjusted ICC: 0.241 
  Unadjusted ICC: 0.241 

LRT for random effect(s):
 Random Effect Dropped  Npar    LL  -2LL   AIC   BIC     LRT DF p.value p.mix
 <none>                    3 -8770 17540 17546 17562                         
 Intercept | SubjectID     2 -8910 17821 17825 17835 280.439  1   <.001 <.001

> print("Model 2: Empty Means, Crossed Random Intercepts for Subjects and Items")
[1] "Model 2: Empty Means, Crossed Random Intercepts for Subjects and Items"
> Model2 = lmer(data = Example6b, REML = TRUE, formula = RT ~ 1 + 
+ (1 | SubjectID) + (1 | ItemID))
> lmerSummary(Model2, ICC1 = TRUE, LRT = TRUE, randCI = TRUE)

Model Fit for Predicting RT:
      AIC       BIC    logLik -2*logLik 
17447.865 17468.819 -8719.933 17439.865 

Model for the Variance:
 Groups     Name       Variance Std.Dev
 ItemID     Intercept  2414.113  49.134
 SubjectID  Intercept  5166.722  71.880
 Residual             14343.043 119.762
Number of obs: 1392 , groups: ItemID, 39; SubjectID, 38 

Fixed Effects:
 Term           Est     SE     DF t.value p.value
 Intercept  635.328 14.434 59.435  44.016   <.001

ICC1 intraclass correlation:
    Adjusted ICC: 0.346 
  Unadjusted ICC: 0.346 

Proportion of Variance by Source (adj = adjusted after fixed effects):
 Source         Variance Prop.Adj Prop.Unadj
 ItemID         2414.113    0.110      0.110
 SubjectID      5166.722    0.236      0.236
 Residual      14343.043    0.654      0.654
 Fixed Effects     0.000               0.000

LRT for random effect(s):
 Random Effect Dropped  Npar    LL  -2LL   AIC   BIC     LRT DF p.value p.mix
 <none>                    4 -8720 17440 17448 17469                         
 Intercept | SubjectID     3 -8879 17758 17764 17780 317.925  1   <.001 <.001
 Intercept | ItemID        3 -8770 17540 17546 17562 100.399  1   <.001 <.001

Random Effect 95% CIs (Fixed Effect +/- 1.96*SQRT(random effect variance)):
 Term                  FixedEffect Variance     SD   Lower   Upper
 Intercept | ItemID        635.328 2414.113 49.134 539.026 731.630
 Intercept | SubjectID     635.328 5166.722 71.880 494.443 776.212

> print("Model 3: Model 2 + Item Frequency, Item Size, and their Interaction")
[1] "Model 3: Model 2 + Item Frequency, Item Size, and their Interaction"
> Model3 = lmer(data = Example6b, REML = TRUE, formula = RT ~ 1 + 
+ freq01 + size01 + freq01:size01 + (1 | SubjectID) + (1 | 
+ ItemID))
> lmerSummary(Model3, LRT = TRUE)

Model Fit for Predicting RT:
      AIC       BIC    logLik -2*logLik 
17416.447 17453.117 -8701.224 17402.447 

Model for the Variance:
 Groups     Name       Variance Std.Dev
 ItemID     Intercept  1692.411  41.139
 SubjectID  Intercept  5168.480  71.892
 Residual             14341.039 119.754
Number of obs: 1392 , groups: ItemID, 39; SubjectID, 38 

Overall F-test for All Fixed Slopes:
 Model  NumDF  DenDF F.value p.value
 Model3     3 31.782   5.343   0.004

Fixed Effects:
 Term               Est     SE     DF t.value p.value
 Intercept      615.783 18.575 60.661  33.151   <.001
 freq01          70.021 20.597 32.401   3.400   0.002
 size01           4.435 20.422 31.392   0.217   0.829
 freq01:size01  -72.031 29.378 31.781  -2.452   0.020

LRT for random effect(s):
 Random Effect Dropped  Npar    LL  -2LL   AIC   BIC     LRT DF p.value p.mix
 <none>                    7 -8701 17402 17416 17453                         
 Intercept | SubjectID     6 -8860 17721 17733 17764 318.202  1   <.001 <.001
 Intercept | ItemID        6 -8733 17466 17478 17510  63.991  1   <.001 <.001

> print("Predicted RT for each Frequency by Size Combination")
[1] "Predicted RT for each Frequency by Size Combination"
> emmSat(Model3, ~freq01 * size01, at = list(freq01 = c(0, 1), 
+ size01 = c(0, 1)))
 freq01 size01  emmean     SE     df lower.CL upper.CL
 0      0      615.783 18.575 60.661  578.636  652.929
 1      0      685.803 18.742 62.663  648.346  723.261
 0      1      620.217 18.549 60.333  583.118  657.317
 1      1      618.208 19.151 58.826  579.884  656.532
Degrees-of-freedom method: satterthwaite
Confidence level used: 0.95

> print("Simple Slopes of Frequency at each Size")
[1] "Simple Slopes of Frequency at each Size"
> FreqSlopes = emtrends(Model3, ~size01, var = "freq01", at = list(size01 = c(0, 
+ 1)), lmer.df = "satterthwaite")
> summary(FreqSlopes, infer = TRUE)
 size01 freq01.trend   SE   df lower.CL upper.CL t.ratio p.value
      0        70.02 20.6 32.4     28.1    112.0   3.400  0.0018
      1        -2.01 20.9 31.2    -44.7     40.7  -0.096  0.9242

Results are averaged over the levels of: freq01 
Degrees-of-freedom method: satterthwaite 
Confidence level used: 0.95 
> print("Simple Slopes of Size at each Frequency")
[1] "Simple Slopes of Size at each Frequency"
> SizeSlopes = emtrends(Model3, ~freq01, var = "size01", at = list(freq01 = c(0, 
+ 1)), lmer.df = "satterthwaite")
> summary(SizeSlopes, infer = TRUE)
 freq01 size01.trend   SE   df lower.CL upper.CL t.ratio p.value
      0         4.43 20.4 31.4    -37.2     46.1   0.217  0.8295
      1       -67.60 21.1 32.1   -110.6    -24.6  -3.201  0.0031

Results are averaged over the levels of: size01 
Degrees-of-freedom method: satterthwaite 
Confidence level used: 0.95 
> PseudoR2(baseModel = Model2, altModel1 = Model3)

Pseudo-R2
 Term                   Model   Variance Pseudo.R2
 Intercept | ItemID     Model2  2414.113          
 Intercept | ItemID     Model3  1692.411     0.299
 Intercept | SubjectID  Model2  5166.722          
 Intercept | SubjectID  Model3  5168.480    -0.000
 Residual               Model2 14343.043          
 Residual               Model3 14341.039     0.000

> TotalR2(altModel1 = Model3)

Total R2
 Model  Total.R2
 Model3    0.030

> print("Model 4a: Model 3 + Random Item Frequency Slope across Subjects")
[1] "Model 4a: Model 3 + Random Item Frequency Slope across Subjects"
> Model4a = lmer(data = Example6b, REML = TRUE, formula = RT ~ 
+ 1 + freq01 + size01 + freq01:size01 + (1 + freq01 | SubjectID) + 
+ 
+ (1 | ItemID))
> lmerSummary(Model4a, LRT = TRUE, randCI = TRUE)

Model Fit for Predicting RT:
      AIC       BIC    logLik -2*logLik 
17415.573 17462.719 -8698.786 17397.573 

Model for the Variance:
 Groups     Name       Variance Std.Dev  Corr
 ItemID     Intercept  1700.404  41.236      
 SubjectID  Intercept  4266.150  65.316      
            freq01      371.707  19.280 0.693
 Residual             14244.399 119.350      
Number of obs: 1392 , groups: ItemID, 39; SubjectID, 38 

Overall F-test for All Fixed Slopes:
 Model   NumDF  DenDF F.value p.value
 Model4a     3 32.512   5.230   0.005

Fixed Effects:
 Term               Est     SE     DF t.value p.value
 Intercept      615.850 17.939 55.823  34.330   <.001
 freq01          69.845 20.860 33.532   3.348   0.002
 size01           4.443 20.448 31.393   0.217   0.829
 freq01:size01  -72.069 29.416 31.784  -2.450   0.020

LRT for random effect(s):
 Random Effect Dropped  Npar    LL  -2LL   AIC   BIC    LRT DF p.value p.mix
 <none>                    9 -8699 17398 17416 17463                        
 freq01 | SubjectID        7 -8701 17402 17416 17453  4.874  2   0.087 0.057
 Intercept | ItemID        8 -8731 17462 17478 17520 64.809  1   <.001 <.001

Random Effect 95% CIs (Fixed Effect +/- 1.96*SQRT(random effect variance)):
 Term                  FixedEffect Variance     SD   Lower   Upper
 Intercept | ItemID        615.850 1700.404 41.236 535.027 696.672
 Intercept | SubjectID     615.850 4266.150 65.316 487.831 743.868
 freq01 | SubjectID         69.845  371.707 19.280  32.057 107.633

> print("Model 4b: Model 3 + Random Item Size Slope across Subjects")
[1] "Model 4b: Model 3 + Random Item Size Slope across Subjects"
> Model4b = lmer(data = Example6b, REML = TRUE, formula = RT ~ 
+ 1 + freq01 + size01 + freq01:size01 + (1 + size01 | SubjectID) + 
+ 
+ (1 | ItemID))
> lmerSummary(Model4b, LRT = TRUE, randCI = TRUE)

Model Fit for Predicting RT:
      AIC       BIC    logLik -2*logLik 
17420.366 17467.512 -8701.183 17402.366 

Model for the Variance:
 Groups     Name       Variance Std.Dev  Corr
 ItemID     Intercept  1692.994  41.146      
 SubjectID  Intercept  5103.813  71.441      
            size01       84.446   9.189 0.067
 Residual             14319.258 119.663      
Number of obs: 1392 , groups: ItemID, 39; SubjectID, 38 

Overall F-test for All Fixed Slopes:
 Model   NumDF  DenDF F.value p.value
 Model4b     3 31.831   5.328   0.004

Fixed Effects:
 Term               Est     SE     DF t.value p.value
 Intercept      615.800 18.529 57.965  33.234   <.001
 freq01          70.001 20.597 32.403   3.399   0.002
 size01           4.415 20.476 31.445   0.216   0.831
 freq01:size01  -72.024 29.378 31.783  -2.452   0.020

LRT for random effect(s):
 Random Effect Dropped  Npar    LL  -2LL   AIC   BIC    LRT DF p.value p.mix
 <none>                    9 -8701 17402 17420 17468                        
 size01 | SubjectID        7 -8701 17402 17416 17453  0.082  2   0.960 0.868
 Intercept | ItemID        8 -8733 17466 17482 17524 64.064  1   <.001 <.001

Random Effect 95% CIs (Fixed Effect +/- 1.96*SQRT(random effect variance)):
 Term                  FixedEffect Variance     SD   Lower   Upper
 Intercept | ItemID        615.800 1692.994 41.146 535.153 696.446
 Intercept | SubjectID     615.800 5103.813 71.441 475.775 755.824
 size01 | SubjectID          4.415   84.446  9.189 -13.596  22.427

